Temperature and disorder chaos in three-dimensional Ising spin glasses

Universal Fragility of Spin Glass Ground States under Single Bond Changes

We consider the effect of perturbing a single bond on ground states of nearest-neighbor Ising spin glasses, with a Gaussian distribution of the coupling constants, across various two- and three-dimensional lattices and regular random graphs. Our results reveal that the ground states are strikingly fragile with respect to such changes. Altering the strength of only a single bond beyond a critical threshold value leads to a new ground state that differs from the original one by a droplet of flipped spins whose boundary and volume diverge with the system size-an effect that is reminiscent of the more familiar phenomenon of disorder chaos. These elementary fractal-boundary zero-energy droplets and their composites feature robust characteristics and provide the lowest-energy macroscopic spin-glass excitations. Remarkably, within numerical accuracy, the size of such droplets conforms to a universal power-law distribution with exponents that depend on the spatial dimension of the system. Furthermore, the critical coupling strengths adhere to a stretched exponential distribution that is predominantly determined by the local coordination number.

Ground-state stability and the nature of the spin glass phase

We propose an approach toward understanding the spin glass phase at zero and low temperature by studying the stability of a spin glass ground state against perturbations of a single coupling. After reviewing the concepts of flexibility, critical droplet, and related quantities for both finite- and infinite-volume ground states, we study some of their properties and review three models in which these quantities are partially or fully understood. We also review a recent result showing the connection between our approach and that of disorder chaos. We then view four proposed scenarios for the low-temperature spin glass phase-replica symmetry breaking, scaling-droplet, TNT, and chaotic pairs-through the lens of the predictions of each scenario for the lowest-energy large-lengthscale excitations above the ground state. Using a new concept called σ-criticality, which quantifies the sensitivity of ground states to single-bond coupling variations, we show that each of these four pictures can be identified with different critical droplet geometries and energies. We also investigate necessary and sufficient conditions for the existence of multiple incongruent ground states.

Learning a local symmetry with neural networks

We explore the capacity of neural networks to detect a symmetry with complex local and non-local patterns: the gauge symmetry Z_{2}. This symmetry is present in physical problems from topological transitions to quantum chromodynamics, and controls the computational hardness of instances of spin-glasses. Here, we show how to design a neural network, and a dataset, able to learn this symmetry and to find compressed latent representations of the gauge orbits. Our method pays special attention to system-wrapping loops, the so-called Polyakov loops, known to be particularly relevant for computational complexity.

From near to eternity: Spin-glass planting, tiling puzzles, and constraint-satisfaction problems

We present a methodology for generating Ising Hamiltonians of tunable complexity and with a priori known ground states based on a decomposition of the model graph into edge-disjoint subgraphs. The idea is illustrated with a spin-glass model defined on a cubic lattice, where subproblems, whose couplers are restricted to the two values {-1,+1}, are specified on unit cubes and are parametrized by their local degeneracy. The construction is shown to be equivalent to a type of three-dimensional constraint-satisfaction problem known as the tiling puzzle. By varying the proportions of subproblem types, the Hamiltonian can span a dramatic range of typical computational complexity, from fairly easy to many orders of magnitude more difficult than prototypical bimodal and Gaussian spin glasses in three space dimensions. We corroborate this behavior via experiments with different algorithms and discuss generalizations and extensions to different types of graphs.

Phase transitions between different spin-glass phases and between different chaoses in quenched random chiral systems

The left-right chiral and ferromagnetic-antiferromagnetic double-spin-glass clock model, with the crucially even number of states q=4 and in three dimensions d=3, has been studied by renormalization-group theory. We find, for the first time to our knowledge, four spin-glass phases, including conventional, chiral, and quadrupolar spin-glass phases, and phase transitions between spin-glass phases. The chaoses, in the different spin-glass phases and in the phase transitions of the spin-glass phases with the other spin-glass phases, with the non-spin-glass ordered phases, and with the disordered phase, are determined and quantified by Lyapunov exponents. It is seen that the chiral spin-glass phase is the most chaotic spin-glass phase. The calculated phase diagram is also otherwise very rich, including regular and temperature-inverted devil's staircases and reentrances.

Devil's staircase continuum in the chiral clock spin glass with competing ferromagnetic-antiferromagnetic and left-right chiral interactions

The chiral clock spin-glass model with q=5 states, with both competing ferromagnetic-antiferromagnetic and left-right chiral frustrations, is studied in d=3 spatial dimensions by renormalization-group theory. The global phase diagram is calculated in temperature, antiferromagnetic bond concentration p, random chirality strength, and right-chirality concentration c. The system has a ferromagnetic phase, a multitude of different chiral phases, a chiral spin-glass phase, and a critical (algebraically) ordered phase. The ferromagnetic and chiral phases accumulate at the disordered phase boundary and form a spectrum of devil's staircases, where different ordered phases characteristically intercede at all scales of phase-diagram space. Shallow and deep reentrances of the disordered phase, bordered by fragments of regular and temperature-inverted devil's staircases, are seen. The extremely rich phase diagrams are presented as continuously and qualitatively changing videos.

Combinatorial optimization using dynamical phase transitions in driven-dissipative systems

The dynamics of driven-dissipative systems is shown to be well-fitted for achieving efficient combinatorial optimization. The proposed method can be applied to solve any combinatorial optimization problem that is equivalent to minimizing an Ising Hamiltonian. Moreover, the dynamics considered can be implemented using various physical systems as it is based on generic dynamics-the normal form of the supercritical pitchfork bifurcation. The computational principle of the proposed method relies on an hybrid analog-digital representation of the binary Ising spins by considering the gradient descent of a Lyapunov function that is the sum of an analog Ising Hamiltonian and archetypal single or double-well potentials. By gradually changing the shape of the latter potentials from a single to double well shape, it can be shown that the first nonzero steady states to become stable are associated with global minima of the Ising Hamiltonian, under the approximation that all analog spins have the same amplitude. In the more general case, the heterogeneity in amplitude between analog spins induces the stabilization of local minima, which reduces the quality of solutions to combinatorial optimization problems. However, we show that the heterogeneity in amplitude can be reduced by setting the parameters of the driving signal near a regime, called the dynamic phase transition, where the analog spins' DC components map more accurately the global minima of the Ising Hamiltonian which, in turn, increases the quality of solutions found. Last, we discuss the possibility of a physical implementation of the proposed method using networks of degenerate optical parametric oscillators.

Chiral Potts spin glass in d=2 and 3 dimensions

The chiral spin-glass Potts system with q=3 states is studied in d=2 and 3 spatial dimensions by renormalization-group theory and the global phase diagrams are calculated in temperature, chirality concentration p, and chirality-breaking concentration c, with determination of phase chaos and phase-boundary chaos. In d=3, the system has ferromagnetic, left-chiral, right-chiral, chiral spin-glass, and disordered phases. The phase boundaries to the ferromagnetic, left- and right-chiral phases show, differently, an unusual, fibrous patchwork (microreentrances) of all four (ferromagnetic, left-chiral, right-chiral, chiral spin-glass) ordered phases, especially in the multicritical region. The chaotic behavior of the interactions, under scale change, are determined in the chiral spin-glass phase and on the boundary between the chiral spin-glass and disordered phases, showing Lyapunov exponents in magnitudes reversed from the usual ferromagnetic-antiferromagnetic spin-glass systems. At low temperatures, the boundaries of the left- and right-chiral phases become thresholded in p and c. In d=2, the chiral spin-glass Potts system does not have a spin-glass phase, consistently with the lower-critical dimension of ferromagnetic-antiferromagnetic spin glasses. The left- and right-chirally ordered phases show reentrance in chirality concentration p.

Zero-temperature quantum annealing bottlenecks in the spin-glass phase

A promising approach to solving hard binary optimization problems is quantum adiabatic annealing in a transverse magnetic field. An instantaneous ground state-initially a symmetric superposition of all possible assignments of N qubits-is closely tracked as it becomes more and more localized near the global minimum of the classical energy. Regions where the energy gap to excited states is small (for instance at the phase transition) are the algorithm's bottlenecks. Here I show how for large problems the complexity becomes dominated by O(log N) bottlenecks inside the spin-glass phase, where the gap scales as a stretched exponential. For smaller N, only the gap at the critical point is relevant, where it scales polynomially, as long as the phase transition is second order. This phenomenon is demonstrated rigorously for the two-pattern Gaussian Hopfield model. Qualitative comparison with the Sherrington-Kirkpatrick model leads to similar conclusions.

Population annealing: Theory and application in spin glasses

Population annealing is an efficient sequential Monte Carlo algorithm for simulating equilibrium states of systems with rough free-energy landscapes. The theory of population annealing is presented, and systematic and statistical errors are discussed. The behavior of the algorithm is studied in the context of large-scale simulations of the three-dimensional Ising spin glass and the performance of the algorithm is compared to parallel tempering. It is found that the two algorithms are similar in efficiency though with different strengths and weaknesses.

Unraveling Quantum Annealers using Classical Hardness

Recent advances in quantum technology have led to the development and manufacturing of experimental programmable quantum annealing optimizers that contain hundreds of quantum bits. These optimizers, commonly referred to as 'D-Wave' chips, promise to solve practical optimization problems potentially faster than conventional 'classical' computers. Attempts to quantify the quantum nature of these chips have been met with both excitement and skepticism but have also brought up numerous fundamental questions pertaining to the distinguishability of experimental quantum annealers from their classical thermal counterparts. Inspired by recent results in spin-glass theory that recognize 'temperature chaos' as the underlying mechanism responsible for the computational intractability of hard optimization problems, we devise a general method to quantify the performance of quantum annealers on optimization problems suffering from varying degrees of temperature chaos: A superior performance of quantum annealers over classical algorithms on these may allude to the role that quantum effects play in providing speedup. We utilize our method to experimentally study the D-Wave Two chip on different temperature-chaotic problems and find, surprisingly, that its performance scales unfavorably as compared to several analogous classical algorithms. We detect, quantify and discuss several purely classical effects that possibly mask the quantum behavior of the chip.

Overfrustrated and underfrustrated spin glasses in d=3 and 2: evolution of phase diagrams and chaos including spin-glass order in d=2

In spin-glass systems, frustration can be adjusted continuously and considerably, without changing the antiferromagnetic bond probability p, by using locally correlated quenched randomness, as we demonstrate here on hypercubic lattices and hierarchical lattices. Such overfrustrated and underfrustrated Ising systems on hierarchical lattices in d=3 and 2 are studied. With the removal of just 51% of frustration, a spin-glass phase occurs in d=2. With the addition of just 33% frustration, the spin-glass phase disappears in d=3. Sequences of 18 different phase diagrams for different levels of frustration are calculated in both dimensions. In general, frustration lowers the spin-glass ordering temperature. At low temperatures, increased frustration favors the spin-glass phase (before it disappears) over the ferromagnetic phase and symmetrically the antiferromagnetic phase. When any amount, including infinitesimal, frustration is introduced, the chaotic rescaling of local interactions occurs in the spin-glass phase. Chaos increases with increasing frustration, as can be seen from the increased positive value of the calculated Lyapunov exponent λ, starting from λ=0 when frustration is absent. The calculated runaway exponent yR of the renormalization-group flows decreases with increasing frustration to yR=0 when the spin-glass phase disappears. From our calculations of entropy and specific-heat curves in d=3, it is shown that frustration lowers in temperature the onset of both long- and short-range order in spin-glass phases, but is more effective on the former. From calculations of the entropy as a function of antiferromagnetic bond concentration p, it is shown that the ground-state and low-temperature entropy already mostly sets in within the ferromagnetic and antiferromagnetic phases, before the spin-glass phase is reached.

Backbone structure of the Edwards-Anderson spin-glass model

We study the ground-state spatial heterogeneities of the Edwards-Anderson spin-glass model with both bimodal and Gaussian bond distributions. We characterize these heterogeneities by using a general definition of bond rigidity, which allows us to classify the bonds of the system into two sets, the backbone and its complement, with very different properties. This generalizes to continuous distributions of bonds the well-known definition of a backbone for discrete bond distributions. By extensive numerical simulations we find that the topological structure of the backbone for a given lattice dimensionality is very similar for both discrete and continuous bond distributions. We then analyze how these heterogeneities influence the equilibrium properties at finite temperature and we discuss the possibility that a suitable backbone picture can be relevant to describe spin-glass phenomena.

Measuring overlaps in mesoscopic spin glasses via conductance fluctuations

We consider the electronic transport in a mesoscopic metallic spin glass. We show that the distribution of overlaps between spin configurations can be inferred from the reduction of the conductance fluctuations by the magnetic impurities. Using this property, we propose new experimental protocols to probe spin glasses directly through their overlaps.

Strong dynamical heterogeneities in the violation of the fluctuation-dissipation theorem in spin glasses

We analyze numerically the violation of the fluctuation-dissipation theorem (FDT) in the +/-J Edwards-Anderson (EA) spin-glass model. Using single spin probability densities we reveal the presence of strong dynamical heterogeneities, which correlate with ground-state information. The physical interpretation of the results shows that the spins can be divided into two sets. In 3D, one set forms a compact structure which presents a coarseninglike behavior with its characteristic violation of the FDT, while the other asymptotically follows the FDT. Finally, we compare the dynamical behavior observed in 3D with 2D.